Parameter Estimation Based on Validity-Aware Gustafson Kessel Clustering

Abstract Regression analysis is the estimation of model parameters using various statistical methods to model the relationship between the dependent variable and one or more independent variables. In classical regression, although all observations in the data set are assumed to have a homogeneous structure, real-life data often have a heterogeneous structure. In these cases where classical methods are inadequate, fuzzy logic-based approaches are used as an alternative to classical methods for estimating the unknown parameters of the regression model. The first step in the process of estimating parameters is to determine the clusters in the data set and to calculate the membership degrees that express the degree of belonging of each observation to these clusters. These membership degrees will be used as weights in the estimation of model parameters of the relevant observations. In this study, since the aim is to perform parameter estimation specifically on ellipsoidal data distributions, the Davies-Bouldin Index (DBI), one of the clustering validity indices that provides effective results in determining the optimal number of clusters, was used, adapted with the Mahalanobis distance. After determining the optimum number of clusters by DBI, the Gustafson-Kessel (GK) clustering algorithm was applied to obtain the membership degrees of the observations. These membership degrees were evaluated as weights that will contribute to the regression model and models were created. To evaluate the performance of the proposed algorithm, applications were made on various data sets and the obtained results were compared with the estimates based on the Fuzzy C-Means (FCM) clustering method.